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Identical expressions
(three x+ two)*ln^3(x)
(3x plus 2) multiply by ln cubed (x)
(three x plus two) multiply by ln cubed (x)
(3x+2)*ln3(x)
3x+2*ln3x
(3x+2)*ln³(x)
(3x+2)*ln to the power of 3(x)
(3x+2)ln^3(x)
(3x+2)ln3(x)
3x+2ln3x
3x+2ln^3x
Similar expressions
(3x-2)*ln^3(x)

The derivative of the function/ (3x+2)*ln^3(x)

Derivative of (3x+2)*ln^3(x)

The solution

You have entered [src]

             3   
(3*x + 2)*log (x)

$$\left(3 x + 2\right) \log{\left(x \right)}^{3}$$

d /             3   \
--\(3*x + 2)*log (x)/
dx

$$\frac{d}{d x} \left(3 x + 2\right) \log{\left(x \right)}^{3}$$

Transform

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = 3 x + 2$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $3 x + 2$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $3$
  2. The derivative of the constant $2$ is zero.
  The result is: $3$
$g{\left(x \right)} = \log{\left(x \right)}^{3}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result of the chain rule is:
  
  $\frac{3 \log{\left(x \right)}^{2}}{x}$
The result is: $3 \log{\left(x \right)}^{3} + \frac{3 \cdot \left(3 x + 2\right) \log{\left(x \right)}^{2}}{x}$
Now simplify:

$\frac{3 \left(x \log{\left(x \right)} + 3 x + 2\right) \log{\left(x \right)}^{2}}{x}$

The answer is:

$\frac{3 \left(x \log{\left(x \right)} + 3 x + 2\right) \log{\left(x \right)}^{2}}{x}$

The graph

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The first derivative [src]

                 2             
     3      3*log (x)*(3*x + 2)
3*log (x) + -------------------
                     x

$$3 \log{\left(x \right)}^{3} + \frac{3 \cdot \left(3 x + 2\right) \log{\left(x \right)}^{2}}{x}$$

Simplify

The second derivative [src]

  /           (-2 + log(x))*(2 + 3*x)\       
3*|6*log(x) - -----------------------|*log(x)
  \                      x           /       
---------------------------------------------
                      x

$$\frac{3 \cdot \left(6 \log{\left(x \right)} - \frac{\left(3 x + 2\right) \left(\log{\left(x \right)} - 2\right)}{x}\right) \log{\left(x \right)}}{x}$$

Simplify

The third derivative [src]

  /                                      /       2              \\
  |                          2*(2 + 3*x)*\1 + log (x) - 3*log(x)/|
3*|-9*(-2 + log(x))*log(x) + ------------------------------------|
  \                                           x                  /
------------------------------------------------------------------
                                 2                                
                                x

$$\frac{3 \left(- 9 \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} + \frac{2 \cdot \left(3 x + 2\right) \left(\log{\left(x \right)}^{2} - 3 \log{\left(x \right)} + 1\right)}{x}\right)}{x^{2}}$$

Simplify

The graph

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Derivative of (3x+2)*ln^3(x)

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The solution